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--- public:image_convolution_in_fourier_domain [2015/12/02 17:29] – created fangfufu
+++ public:image_convolution_in_fourier_domain [2018/03/31 00:38] (current) – external edit 127.0.0.1
@@ Line 1: / Line 1: @@
+====== Image convolution and deconvolution in the frequency domain ======
+Image convolution and deconvolution can be performed quickly in the frequency domain. In the frequency domain, pointwise multiplication is convolution, while pointwise division is deconvolution. The pseudo-code of the procedure is:
+  - Convert your input image into the frequency domain, via fast Fourier transform.
+  - Pad your convolution kernel with zeros, until it reaches the same size as your input image.
+  - Convert your convolution kernel into the frequency domain, via fast Fourier transform.
+  - Pointwise multiply the frequency domain input image and the frequency domain convolution kernel together.
+  - Convert the product from pointwise multiplication back into the spatial domain via inverse fast Fourier transform.
+If you want to perform deconvolution, rather than perform pointwise multiplication in step 4, you do pointwise division. You have to be aware divide-by-zero problem, when you are performing deconvolution. You might want to replace all the zeros with eps().
+Below are some Matlab examples. Note that Matlab takes care of the zero-padding inside the fft2() function call.
+===== Computing the Laplacian of an image via convolution =====
+<code matlab>
+function [ out ] = get_laplacian( in )
+%GET_LAPLACIAN Obtain the Laplacian of an image
+% This is an implementation of the algorithm described in page 32-35 of
+% Lightness and Brightness Computation by Retinex-like Algorithms by Eran
+% Borenstein
+if size(in, 3) == 3
+    for i = 1:3
+        tmp = in(:,:,i);
+        out(:,:,i) = get_laplacian_real(tmp);
+    end
+else
+    out = get_laplacian_real(in);
+end
+end
+function [ out ] = get_laplacian_real(in)
+%% Fourier transform input image
+a = fft2(in);
+b = fft2(fspecial('laplacian'), size(in, 1), size(in, 2));
+c = a.*b;
+out = real(ifft2(c));
+end
+</code>
+===== Computing the Laplacian of an image via deconvolution =====
+<code matlab>
+function [ out ] = inv_laplacian(in)
+%INV_LAPLACIAN Convert a Laplacian image back to its original
+if size(in, 3) == 3
+    for i = 1:3
+        tmp = in(:,:,i);
+        out(:,:,i) = inv_laplacian_real(tmp);
+    end
+else
+    out = inv_laplacian_real(in);
+end
+end
+function [out] = inv_laplacian_real(in)
+%% Fourier transform input image
+a = fft2(in);
+b = fft2(fspecial('laplacian'), size(in, 1), size(in, 2));
+b(b == 0) = eps;
+c = a./b;
+out = real(ifft2(c));
+out = out + abs(min(out(:)));
+end
+</code>